## Is Arrow’s Theorem interesting?

Suppose a group of people has to make a choice from a set S of options.  Each member of the group ranks the options in S from best to worst.  A “voting system” is a mechanism for aggregating these rankings into a single ranking, meant to represent the preferences of the group as a whole.

There are certain natural features you’d like a voting system to have.  For instance, you might want it to be “monotone” — if a voter who likes option A better than B switches those two in her ranking, that shouldn’t improve A’s overall position or worsen B’s.

Kenneth Arrow wrote down a modest list of axioms, including monotonicity, that seem like pretty non-negotiable features you’d want a voting system to have.  Then he proved that no voting system satisfies all the axioms when S consists of more than two options.

Why wouldn’t that be interesting?

Well, here are some axioms that are not on Arrow’s list:

• Anonymity (the overall outcome is invariant under permutation of voters)
• Neutrality (the overall outcome is invariant under permutation of options)

Surely you don’t really want to consider a voting system that doesn’t meet these requirements.  But if you add these two requirements, the resulting special case of Arrow’s theorem was proved more than 150 years earlier, by Condorcet!  Namely:  it is not hard to check that when |S| = 2, the only anonymous, neutral, Arrovian voting system is majority rule.  Add to that Arrow’s axiom of  “independence of irrelevant alternatives” and you get

(*) if a majority of the population ranks A above B, then A must finish above B in the final ranking.

But what Condorcet observed is the following discomfiting phenomenon:  suppose there are three options, and suppose that the rankings in the population are equally divided between A>B>C, C>A>B, and B>C>A.  Then a majority ranks A over B, a majority ranks B over C, and a majority ranks C over A.  This contradicts (*).

Given this, my question is:  why is Arrow’s theorem considered such a big deal in the theory of social choice?  Suppose it were false, and there were a non-anonymous or non-neutral voting mechanism that satisfied Arrow’s other axioms; would there be any serious argument that such a voting system should be adopted?

Thanks to Greg Kuperberg for some helpful explanation about this stuff on Google+.  Relevant reading:  Ben Webster says Arrow’s Theorem is a scam, but not for the reasons discussed in this post.

## Condorcet on French mathematics as underdog

Quite striking and strange for a modern mathematician to read the following, from Condorcet’s 1787  lectures to the Lycee:

It is to French mathematicians that we owe the theory of probability calculus.  This is perhaps worth saying.  Other nations, and often even Frenchmen themselves, have reproached us for lacking the gift of invention, granting us only the ability to perfect other people’s discoveries…

Tagged , ,

## Condorcet was an interesting dude

I knew about him only in relation with the voting paradox.  But he also wrote The Future Progress of the Human Mind (1795), a utopian tract featuring surprsingly modern stuff like this:

No one has ever believed that the human mind could exhaust all the facts of nature, all the refinements of measuring and analyzing these facts, the inter relationship of objects, and all the possible combinations of ideas….

But because, as the number of facts known increases, man learns to classify them, to reduce them to more general terms; because the instruments and the methods of observation and exact measurement are at the same time reaching a new precision; . . . the truths whose discovery has cost the most effort, which at first could be grasped only by men capable of profound thought, are soon carried further and proved by methods that are no longer beyond the reach of ordinary intelligence. If the methods that lead to new combinations are exhausted, if their application to problems not yet solved requires labors that exceed the time or the capacity of scholars, soon more general methods, simpler means, come to open a new avenue for genius….

The organic perfectibility or degeneration of races in plants and animals may be regarded as one of the general laws of nature.

This law extends to the human species; and certainly no one will doubt that progress in medical conservation [of life], in the use of healthier food and housing, a way of living that would develop strength through exercise without impairing it by excess, and finally the destruction of the two most active causes of degradation-misery and too great wealth-will prolong the extent of life and assure people more constant health as well as a more robust constitution. We feel that the progress of preventive medicine as a preservative, made more effective by the progress of reason and social order, will eventually banish communicable or contagious illnesses and those diseases in general that originate in climate, food, and the nature of work. It would not be difficult to prove that this hope should extend to almost all other diseases, whose more remote causes will eventually be recognized. Would it be absurd now to suppose that the improvement of the human race should be regarded as capable of unlimited progress? That a time will come when death would result only from extraordinary accidents or the more and more gradual wearing out of vitality, and that, finally, the duration of the average interval between birth and wearing out has itself no specific limit whatsoever? No doubt man will not become immortal, but cannot the span constantly increase between the moment he begins to live and the time when naturally, without illness or accident, he finds life a burden?

The voting paradoxes are found in Condorcet’s 1785 treatise Essay on the Application of Analysis to the Probability of Majority Decisions. But the main body of the book isn’t about voting paradoxes; it’s an attempt to provide mathematical backing for democratic theory.  Condorcet argued that the probability of the majority holding the wrong position was much smaller than the chance that the minority would be in the wrong.  So democracy is justified not only on principle, but because it is more likely to yield true beliefs on the part of the government.  I learned this, and other interesting facts, from Trevor Pateman’s article “Majoritarianism,” which presents Condorcet as a kind of quantitative version of Rousseau.

Tagged , , , ,